Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $z = \dfrac{36a + 9}{4} \div \dfrac{a(4a + 1)}{-4} $
Solution: Dividing by an expression is the same as multiplying by its inverse. $z = \dfrac{36a + 9}{4} \times \dfrac{-4}{a(4a + 1)} $ When multiplying fractions, we multiply the numerators and the denominators. $z = \dfrac{ (36a + 9) \times -4 } { 4 \times a(4a + 1) } $ $ z = \dfrac {-4 \times 9(4a + 1)} {4 \times a(4a + 1)} $ $ z = \dfrac{-36(4a + 1)}{4a(4a + 1)} $ We can cancel the $4a + 1$ so long as $4a + 1 \neq 0$ Therefore $a \neq -\dfrac{1}{4}$ $z = \dfrac{-36 \cancel{(4a + 1})}{4a \cancel{(4a + 1)}} = -\dfrac{36}{4a} = -\dfrac{9}{a} $